Defining Gm and Yoneda and group objects
... between the Z-points of X and Y for each scheme Z and these maps satisfy certain naturality conditions, then they arise from a map from X to Y . This correspondence also helps us make sense of a group object in a category, which is an object G and morphisms m : G × G → G, i : G → G and e : ? → G, wh ...
... between the Z-points of X and Y for each scheme Z and these maps satisfy certain naturality conditions, then they arise from a map from X to Y . This correspondence also helps us make sense of a group object in a category, which is an object G and morphisms m : G × G → G, i : G → G and e : ? → G, wh ...
AAG, LECTURE 13 If 0 → F 1 → F2 → F3 → 0 is a short exact
... but it is not necessarily the case that F2 (X) → F3 (X) is surjective. (The surjectivity of F2 → F3 implies something weaker: that for any f ∈ F3 (X) and x ∈ X there is an open x ∈ U ⊂ X and a g ∈ F2 (U ) so that g 7→ f|U . These g’s are not unique, since we can change them by sections of F1 , so on ...
... but it is not necessarily the case that F2 (X) → F3 (X) is surjective. (The surjectivity of F2 → F3 implies something weaker: that for any f ∈ F3 (X) and x ∈ X there is an open x ∈ U ⊂ X and a g ∈ F2 (U ) so that g 7→ f|U . These g’s are not unique, since we can change them by sections of F1 , so on ...
THE BRAUER GROUP 0.1. Number theory. Let X be a Q
... • (Wedderburn) B( n ) = 0, B(K) = 0 for any K which is an algebraic extension of a finite field. • (Frobenius) B( ) ' /2 (representatives: , ) • B( ) = 0, B(k) = 0 for any (separably) closed field k. • (Tsen-Lang) B( [x]) = 0, B(k(X)) = 0 for any curve X defined over an algebraically closed field. ...
... • (Wedderburn) B( n ) = 0, B(K) = 0 for any K which is an algebraic extension of a finite field. • (Frobenius) B( ) ' /2 (representatives: , ) • B( ) = 0, B(k) = 0 for any (separably) closed field k. • (Tsen-Lang) B( [x]) = 0, B(k(X)) = 0 for any curve X defined over an algebraically closed field. ...
Classifying spaces and spectral sequences
... present popularization will be of some interest. Apart from this my purpose is to obtain for a generalized cohomology theory k* a spectral sequence connecting A*(X) with the ordinary cohomology of X. This has been done in the past [i], when X is a GW-complex, by considering the filtration ofX by its ...
... present popularization will be of some interest. Apart from this my purpose is to obtain for a generalized cohomology theory k* a spectral sequence connecting A*(X) with the ordinary cohomology of X. This has been done in the past [i], when X is a GW-complex, by considering the filtration ofX by its ...
A BRIEF INTRODUCTION TO SHEAVES References 1. Presheaves
... A ringed space (X, O) is a local ringed space if each stalk Ox (x ∈ X) is a local ring with maximal ideal mX ; a morphism (X, OX ) −→ (Y, OY ) of local ringed spaces is one where on each stalk OY,f (x) −→ f∗ OX,x is a homomorphism of local rings (i.e., a ring homomorphism h : R −→ S for which h−1 mS ...
... A ringed space (X, O) is a local ringed space if each stalk Ox (x ∈ X) is a local ring with maximal ideal mX ; a morphism (X, OX ) −→ (Y, OY ) of local ringed spaces is one where on each stalk OY,f (x) −→ f∗ OX,x is a homomorphism of local rings (i.e., a ring homomorphism h : R −→ S for which h−1 mS ...
LECTURE 21 - SHEAF THEORY II 1. Stalks
... PIOTR MACIAK Abstract. This lecture develops the ideas introduced in Lecture 20. In particular, we define a stalk of a sheaf and use this concept along with the concept of a sheaf of continuous sections to show that there is a natural way to associate a sheaf to every presheaf. ...
... PIOTR MACIAK Abstract. This lecture develops the ideas introduced in Lecture 20. In particular, we define a stalk of a sheaf and use this concept along with the concept of a sheaf of continuous sections to show that there is a natural way to associate a sheaf to every presheaf. ...
Algebraic Geometry
... geometric object to every commutative ring. Unfortunately, A 7! spm A is not functorial in this generality: if 'W A ! B is a homomorphism of rings, then ' 1 .m/ for m maximal need not be maximal — consider for example the inclusion Z ,! Q. Thus he was forced to replace spm.A/ with spec.A/, the set o ...
... geometric object to every commutative ring. Unfortunately, A 7! spm A is not functorial in this generality: if 'W A ! B is a homomorphism of rings, then ' 1 .m/ for m maximal need not be maximal — consider for example the inclusion Z ,! Q. Thus he was forced to replace spm.A/ with spec.A/, the set o ...
PH Kropholler Olympia Talelli
... of the coinduced module CoindyH and contains G-fixed points in the form of the constant functions. The action of G is given by @(g’) = 4( gg’). We prove that Z is ZF-free for each finite F using a result of Nobeling that [s2, Z] is free abelian for any set 0; see [ 1, Corollary 97.41. Let F be a fin ...
... of the coinduced module CoindyH and contains G-fixed points in the form of the constant functions. The action of G is given by @(g’) = 4( gg’). We prove that Z is ZF-free for each finite F using a result of Nobeling that [s2, Z] is free abelian for any set 0; see [ 1, Corollary 97.41. Let F be a fin ...
Notes
... Proof. We construct a quasi-inverse. Let S be a continuous G-set. Let s 2 S be a point. Write H for the stabiliser of s. It is a open subgroup. By elementary group theory, we have an isomorphism of G-sets G=H ! Gs. This shows that S is the disjoint union of nite orbits. We map Gs to Spec(kH ), and ...
... Proof. We construct a quasi-inverse. Let S be a continuous G-set. Let s 2 S be a point. Write H for the stabiliser of s. It is a open subgroup. By elementary group theory, we have an isomorphism of G-sets G=H ! Gs. This shows that S is the disjoint union of nite orbits. We map Gs to Spec(kH ), and ...
THE GEOMETRY OF FORMAL VARIETIES IN ALGEBRAIC
... Shifting gears slightly, one of the classical presentations of infinitesimal elements are elements that “square to zero.” This turns out to be a fairly difficult thing to make both precise and analytically useful, which is why most analysts spend their time thinking about other things — but it’s a v ...
... Shifting gears slightly, one of the classical presentations of infinitesimal elements are elements that “square to zero.” This turns out to be a fairly difficult thing to make both precise and analytically useful, which is why most analysts spend their time thinking about other things — but it’s a v ...
Sheaf Cohomology 1. Computing by acyclic resolutions
... For the particular functor ‘take global sections’, other conditions on a sheaf still guarantee acyclicity and at the same time are ‘intrinsic’ in that they do not refer to any ‘ambient category’. Threfore, in principle these other conditions are more readily verifiable. For the moment, we merely cat ...
... For the particular functor ‘take global sections’, other conditions on a sheaf still guarantee acyclicity and at the same time are ‘intrinsic’ in that they do not refer to any ‘ambient category’. Threfore, in principle these other conditions are more readily verifiable. For the moment, we merely cat ...
Cohomology jump loci of quasi-projective varieties Botong Wang June 27 2013
... Here the isomorphism is an isomorphism of real Lie groups, not complex Lie groups. However, we will not use this at all in this talk. Instead, we need to introduce our main player, the cohomology jump loci in MB (X ), def ...
... Here the isomorphism is an isomorphism of real Lie groups, not complex Lie groups. However, we will not use this at all in this talk. Instead, we need to introduce our main player, the cohomology jump loci in MB (X ), def ...
Section 07
... where the product ranges over all (n + 1)-tuples i0 , . . . , in for which Ui0 ...in is non-empty. This is a larger complex with much redundant information (the group A(Ui0 ...in ) now occurs (n + 1)! times), hence less convenient for calculations, but it computes the same cohomology as we will now ...
... where the product ranges over all (n + 1)-tuples i0 , . . . , in for which Ui0 ...in is non-empty. This is a larger complex with much redundant information (the group A(Ui0 ...in ) now occurs (n + 1)! times), hence less convenient for calculations, but it computes the same cohomology as we will now ...
Generalities About Sheaves - Lehrstuhl B für Mathematik
... ker(ϕ) is a subsheaf of F. ϕ is injective ⇐⇒ ker(ϕ) = 0. The sheaf im(ϕ) is im(ϕ) := impresheaf (ϕ)+ ϕ is surjective ⇐⇒ im(ϕ) = G. ϕ is surjective ⇐⇒ ∀x ∈ X ...
... ker(ϕ) is a subsheaf of F. ϕ is injective ⇐⇒ ker(ϕ) = 0. The sheaf im(ϕ) is im(ϕ) := impresheaf (ϕ)+ ϕ is surjective ⇐⇒ im(ϕ) = G. ϕ is surjective ⇐⇒ ∀x ∈ X ...
RIGID RATIONAL HOMOTOPY THEORY AND
... Suppose that k is a finite field, and X{k is a geometrically connected variety (“ separated scheme of finite type). Question. Is there a way to ’do algebraic topology on X’ ? More specifically, we can ask if there are cohomology functors H i pXq which is some way behave like the singular cohomology ...
... Suppose that k is a finite field, and X{k is a geometrically connected variety (“ separated scheme of finite type). Question. Is there a way to ’do algebraic topology on X’ ? More specifically, we can ask if there are cohomology functors H i pXq which is some way behave like the singular cohomology ...
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... Definition 2. A subset A of an irreducible variety V /K is said to be a thin algebraic set (or thin set, or “mince” set) if it is a union of a finite number of subsets of type C1 and type C2 . ...
... Definition 2. A subset A of an irreducible variety V /K is said to be a thin algebraic set (or thin set, or “mince” set) if it is a union of a finite number of subsets of type C1 and type C2 . ...
ON TAMAGAWA NUMBERS 1. Adele geometry Let X be an
... The adele geometry is the study of the pair (XA, XQ), together with the imbedding above. Therefore, one has to define all conceivable invariants of X in terms of the pair and study relations among them or connections with other invariants of X. The Tamagawa number x (X) is an example of such invaria ...
... The adele geometry is the study of the pair (XA, XQ), together with the imbedding above. Therefore, one has to define all conceivable invariants of X in terms of the pair and study relations among them or connections with other invariants of X. The Tamagawa number x (X) is an example of such invaria ...
Complex Bordism (Lecture 5)
... resulting elements of E 0 ({x}) are off by a sign if we choose trivializations with different orientations. Remark 4. A consequence of Definition 2 is that the Leray-Serre spectral sequence for the fibration S(ζ) → X degenerates and gives an identification E ∗ (X) ' E ∗+n−ζ (X), given by multiplicat ...
... resulting elements of E 0 ({x}) are off by a sign if we choose trivializations with different orientations. Remark 4. A consequence of Definition 2 is that the Leray-Serre spectral sequence for the fibration S(ζ) → X degenerates and gives an identification E ∗ (X) ' E ∗+n−ζ (X), given by multiplicat ...
NOTES ON GROTHENDIECK TOPOLOGIES 1
... The canonical topology on TG turns out to be the same as the one defined earlier. So U 7→ Hom(U, Z ) are all sheaves. It turns out these are all the sheaves of sets on TG . Proposition 4.2. There is an equivalence between the category of left G-sets and the category of sheaves of sets on TG . The fu ...
... The canonical topology on TG turns out to be the same as the one defined earlier. So U 7→ Hom(U, Z ) are all sheaves. It turns out these are all the sheaves of sets on TG . Proposition 4.2. There is an equivalence between the category of left G-sets and the category of sheaves of sets on TG . The fu ...
Homework Set 3 Solutions are due Monday, November 9th.
... corresponds to the map induced by inclusion F → F 0 . Problem 5. Deduce from the previous problem that if f : F → G is a morphism of sheaves of abelian groups on X, then Im(f ) is canonically isomorphic to the subsheaf F 0 of G, where F 0 (U ) consists of those s ∈ G(U ) such that for all x ∈ X, the ...
... corresponds to the map induced by inclusion F → F 0 . Problem 5. Deduce from the previous problem that if f : F → G is a morphism of sheaves of abelian groups on X, then Im(f ) is canonically isomorphic to the subsheaf F 0 of G, where F 0 (U ) consists of those s ∈ G(U ) such that for all x ∈ X, the ...
LECTURE NOTES 4: CECH COHOMOLOGY 1
... for q > 0. Our work on double complexes now easily implies the desired result. 8. Proof of Proposition 8.1 The comparison of singular and Cech cohomology depended on the following result. Let U be a cover of a space X. Let A be any abelian group, and let S ∗ be the presheaf on X given by the formula ...
... for q > 0. Our work on double complexes now easily implies the desired result. 8. Proof of Proposition 8.1 The comparison of singular and Cech cohomology depended on the following result. Let U be a cover of a space X. Let A be any abelian group, and let S ∗ be the presheaf on X given by the formula ...
1. Let G be a sheaf of abelian groups on a topological space. In this
... 1. Let G be a sheaf of abelian groups on a topological space. In this problem, we define H 1 (X, G) as the set of isomorphism classes of G-torsors on X. Let F be a G-torsor, and F 0 be a sheaf of sets with an action by G. Recall that we defined F ×G F 0 as the sheafification of the presheaf that ass ...
... 1. Let G be a sheaf of abelian groups on a topological space. In this problem, we define H 1 (X, G) as the set of isomorphism classes of G-torsors on X. Let F be a G-torsor, and F 0 be a sheaf of sets with an action by G. Recall that we defined F ×G F 0 as the sheafification of the presheaf that ass ...
Introduction to Sheaves
... 1. The Constant Presheaf; Let G be a abelian group and let F be the contravarient function from open sets of X to abeilan groups, such that F (U ) = G. 2. Real valued functions; Let O(U ) denote all functions f : U → R. These functions form a group under pointwise addition, and give the structure of ...
... 1. The Constant Presheaf; Let G be a abelian group and let F be the contravarient function from open sets of X to abeilan groups, such that F (U ) = G. 2. Real valued functions; Let O(U ) denote all functions f : U → R. These functions form a group under pointwise addition, and give the structure of ...
0.1 A lemma of Kempf
... Theorem 1 (Serre). Let X = SpecA be an affine scheme, F a quasi-coherent sheaf. Then H i (X, F) = 0 for i ≥ 1. We shall prove this result following [?]. The idea is that X has a very nice basis: namely, the family of all sets D(f ), f ∈ A. These are themselves affine, and moreover the intersection o ...
... Theorem 1 (Serre). Let X = SpecA be an affine scheme, F a quasi-coherent sheaf. Then H i (X, F) = 0 for i ≥ 1. We shall prove this result following [?]. The idea is that X has a very nice basis: namely, the family of all sets D(f ), f ∈ A. These are themselves affine, and moreover the intersection o ...